Constructing Versatile Constructing Versatile Mathematical ConceptionsMathematical Conceptions

Mike Thomas

The University of Auckland

OverviewOverview

Define versatile thinking in mathematics

Consider some examples and problems

Versatile thinkingVersatile thinking in mathematics in mathematics

First…process/object versatility—the ability to switch at will in any given representational system between a perception of symbols as a process or an object

Not just procepts, which are arithmetic/algebraic

Lack of process-object versatilityLack of process-object versatility

(Thomas, 1988; 2008)

Visuo/analytic versatilityVisuo/analytic versatility

Visuo/analytic versatility—the ability to exploit the power of visual schemas by linking them to relevant logico/analytic schemas

A Model of Cognitive IntegrationA Model of Cognitive Integration

Higher level schemas

Lower level schemas

C–links andA–links

Directed

conscious

unconscious

Representational VersatilityRepresentational Versatility

Thirdly…

representational versatility—the ability to work seamlessly within and between representations, and to engage in procedural and conceptual interactions with representations

Treatment and conversionTreatment and conversion

Duval, 2006, p. 3

Icon to symbol requires interpretation through Icon to symbol requires interpretation through appropriate mathematical schema to ascertain appropriate mathematical schema to ascertain propertiesproperties

External world

Internal world

external sign

‘appropriate’ schema

interpretInteract

with/act on

translation or conversion

ExampleExample

This may be an icon, This may be an icon, a ‘hill’, saya ‘hill’, say

We may look We may look ‘deeper’ and see a ‘deeper’ and see a parabola using a parabola using a quadratic function quadratic function schemaschema

This schema may allow us to convert to algebra

A possible problemA possible problem

The opportunity to acquire knowledge in a variety of forms, and to establish connections between different forms of knowledge are apt to contribute to the flexibility of students’ thinking (Dreyfus and Eisenberg, 1996). The same variety, however, also tends to blur students’ appreciation of the difference in status which different means of establishing mathematical knowledge bestow upon that knowledge.

Dreyfus, 1999

Algebraic symbols: Equals schemaAlgebraic symbols: Equals schema

• Pick out those statements that are equations from the following list and write down why you think the statement is an equation:

• a) k = 5• b) 7w – w• c) 5t – t = 4t• d) 5r – 1 = –11• e) 3w = 7w – 4w

Equation schema: only needs an Equation schema: only needs an operationoperation

Perform an operation and get a result:Perform an operation and get a result:

Another possible problemAnother possible problem

Compartmentalization “This phenomenon occurs when a person has

two different, potentially conflicting schemes in his or her cognitive structure. Certain situations stimulate one scheme, and other situations stimulate the other…Sometimes, a given situation does not stimulate the scheme that is the most relevant to the situation. Instead, a less relevant scheme is activated”

Vinner & Dreyfus, 1989, p. 357

A formulaA formula

Linking of representation Linking of representation systems systems (x, 2x), where x is a real number

Ordered pairs to algebra to graph

Abstraction in contextAbstraction in context

We also pay careful attention to the multifaceted context in which processes of abstraction occur: A process of abstraction is influenced by the task(s) on which students work; it may capitalize on tools and other artifacts; it depends on the personal histories of students and teacher

Hershkowitz, Schwarz, Dreyfus, 2001

Abstraction of meaning for Abstraction of meaning for

Expression Rate ofchange

Gradientof

tangent

Derivative Term inan

equation

d y

d x

= 5 x 16 6 11 2

2 x +

d y

d x

= 1 3 0 5 8

d y

d x

= 4 y 7 4 7 1

z =

d (

d y

d x

)

d x

1 1 0 3

dy

dx

Process/object versatility for Process/object versatility for

Seeing solely as a process causes a

problem interpreting

and relating it to

d(dydx)dx

d2ydx2

dy

dx

dy

dx

Student: that does imply the second derivative…it is the derived function of the second derived function

))(( xff ′′

To see this relationship “one needs to deal with the function g as an object that is operated on in two ways”

Dreyfus (1991, p. 29)

g(x)dx =a+k

b+k

∫ g(x−k)dxa

b

∫

Proceptual versatile thinkingProceptual versatile thinking

If ,

then write down the value of

€

f (t)dt = 8.61

3

∫

€

f (t −1)dt2

4

∫

Versatile thinking–change of Versatile thinking–change of representation systemrepresentation system

nb The representation does not correspond; an exemplar y=x2 is used

Newton-Raphson versatilityNewton-Raphson versatility

x2 =x1 −f(x1)′f (x1)

Many students can use the formula below to calculate a better approximation of the root, but are unable to explain why it works

Newton-RaphsonNewton-Raphson

x3 x2 x1

f(x) f(x1)

′f (x1) =f (x1)

x1 − x2

Newton-RaphsonNewton-Raphson

When is x1 a suitable first approximation for the root a of f(x) = 0?

Student V1: It is very important that the approximation is close enough the root and not on a turning point. Otherwise you might be finding the wrong root.

Student knowledge constructionStudent knowledge construction

Learning may take place in a single representation system, so inter-representational links are not made

Avoid activity comprising surface interactions with a representation, not leading to the concept

The same representations may mean different things to students due to their contextual schema construction (abstraction)

Use multiple contexts for representations

ReferenceReference

Thomas, M. O. J. (2008). Developing versatility in mathematical thinking. Mediterranean Journal for Research in Mathematics Education, 7(2), 67-87.

From: moj.thomas@auckland.ac.nz

Proceptual versatility–Proceptual versatility–eigenvectorseigenvectors

€

Ax = λxTwo different processes

Need to see resulting object or ‘effect’ as the same

Work within the representation Work within the representation system—algebrasystem—algebra

Work within the representation Work within the representation system—algebrasystem—algebra

Same process

ConversionConversion

v

v

u

u

Student knowledge constructionStudent knowledge construction

Learning may take place in a single representation system, so inter-representational links are not made

Avoid activity comprising surface interactions with a representation, not leading to the concept

The same representations may mean different things to students due to their contextual schema construction (abstraction)

Use multiple contexts for representations

ConversionsConversions

Translation between registers

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Duval, 1999, p. 5

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

RepresentationsRepresentations

Duval, 1999, p. 4

Semiotic representations systems — Semiotic registers

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Processing — transformation within a register

Epistemic actions are mental actions by means of which knowledge is used or constructed

Representations and Representations and mathematicsmathematics Much of mathematics is about what we can

learn about concepts through their representations (or signs)

Examples include: natural language, algebras, graphs, diagrams, pictures, sets, ordered pairs, tables, presentations, matrices, etc. (nb icons, indices and symbols here)

Some of the things we learn are representation dependant; others representation independent

Representation dependant ideas...Representation dependant ideas...

"…much of the actual work of mathematics is to determine exactly what structure is preserved in that representation.”

J. KaputIs 12 even or odd? Numbers ending in a multiple of 2 are even. True or False?123?

123, 345, 569 are all odd numbers113, 346, 537, 469 are all even numbers

Representational interactionsRepresentational interactions

We can interact with a representation by:

Observation—surface or deep (property)

Performing an action—procedural or conceptual

Thomas, 2001

Representational interactionsRepresentational interactions

We can interact with a representation by:

Observation—surface or deep (property)

Performing an action—procedural or conceptual

Thomas, 2001

€

f (x ) =x 2

x 2 −1

Let

For what values of x is f(x) increasing?

Some could answer this using algebra and but…

Procedure versus conceptProcedure versus concept

′f (x) > 0

Procedure versus conceptProcedure versus concept

0.50 1.00 1.50 2.00 2.50 3.00-0.50-1.00-1.50-2.00-2.50

-1.00

-2.00

-3.00

1.00

2.00

3.00

4.00

For what values of x is this function increasing?

Why it may failWhy it may fail

We should not think that the three parts of versatile thinking are independent

Neither should we think that a given sign has a single interpretation — it is influenced by the context

Icon, index, symbolIcon, index, symbol

A

D C

B

ABCD — symbol

Icon to symbolIcon to symbol

“”

Moving from seeing a drawing (icon) to seeing a figure (symbol) requires interpretation; use of an overlay of an appropriate mathematical schema to ascertain properties

Function